Question

question: 9
serenity uses the following polynomial expression x(x+5)(x-7) to model the volume of a rectangular prism. the height is shorter than the width and the length. what is the height of this prism?

a. x

b. x-7

c. x+5

d. x³ - 2x² -35x

Explanation:

Step1: Analyze the polynomial factors

The volume of a rectangular prism is given by \( V = l \times w \times h \), where \( l \) is length, \( w \) is width, and \( h \) is height. The polynomial is \( x(x + 5)(x - 7) \), so the three factors are \( x \), \( x + 5 \), and \( x - 7 \).

Step2: Compare the factors' sizes

We need to determine which factor is the smallest (since height is shorter than width and length). Let's analyze the expressions:
- For \( x \): It represents a linear term with coefficient 1.
- For \( x + 5 \): This is \( x \) plus 5, so it's larger than \( x \) for positive \( x \) (since volume dimensions are positive, \( x>7 \) to make \( x - 7 \) positive).
- For \( x - 7 \): Wait, no—if \( x>7 \), \( x - 7 \) is positive, but let's check the size relative to \( x \). Wait, no, actually, we made a mistake. Wait, \( x - 7 \) would be smaller than \( x \) (since we subtract 7), but wait, no—wait, the height has to be a positive length, so \( x - 7>0 \implies x>7 \). Then, \( x - 7 \) vs \( x \): \( x - 7 < x \), but \( x + 5>x \). But wait, the problem says "height is shorter than width and length". Wait, but \( x - 7 \) could be negative if \( x<7 \), which is not possible for a dimension. So we must have \( x>7 \), so \( x - 7 \) is positive, \( x \) is positive, \( x + 5 \) is positive. Now, compare the three: \( x - 7 \), \( x \), \( x + 5 \). The smallest among them is \( x - 7 \)? Wait, no—wait, \( x - 7 = x - 7 \), \( x = x \), \( x + 5 = x + 5 \). So \( x - 7 < x < x + 5 \) when \( x>7 \). Wait, but that would mean \( x - 7 \) is the smallest. But wait, the options are A. \( x \), B. \( x - 7 \), C. \( x + 5 \), D. the expanded form. Wait, but maybe we made a mistake. Wait, no—wait, the height has to be a positive length, so \( x - 7>0 \implies x>7 \). Then, \( x - 7 \) is less than \( x \), which is less than \( x + 5 \). But wait, the problem says "height is shorter than width and length". So the smallest factor is \( x - 7 \)? Wait, but let's check again. Wait, maybe the error is in assuming \( x - 7 \) is positive. Wait, no—volume can't have negative dimensions, so \( x - 7>0 \implies x>7 \). So \( x - 7 \) is positive, \( x \) is positive, \( x + 5 \) is positive. Now, the three factors: \( x - 7 \), \( x \), \( x + 5 \). The order from smallest to largest is \( x - 7 < x < x + 5 \). So the smallest is \( x - 7 \), which would be the height. Wait, but let's check the options. Option B is \( x - 7 \), A is \( x \), C is \( x + 5 \), D is the expanded polynomial (which is the volume, not a dimension). So the height should be the smallest factor, which is \( x - 7 \)? Wait, but wait, no—wait, maybe we messed up. Wait, no, let's re-express: the volume is \( x(x + 5)(x - 7) \), so the three dimensions are \( x \), \( x + 5 \), and \( x - 7 \). Since height is shorter than width and length, we need the smallest of these three. Let's take \( x = 8 \) (since \( x>7 \) to make \( x - 7 \) positive). Then \( x - 7 = 1 \), \( x = 8 \), \( x + 5 = 13 \). So the smallest is \( 1 \) (which is \( x - 7 \) when \( x = 8 \)), so the height is \( x - 7 \). Wait, but the option B is \( x - 7 \). But wait, the initial thought was wrong—we thought \( x \) was the smallest, but no, \( x - 7 \) is smaller when \( x>7 \). Wait, but let's check the problem again. The problem says "height is shorter than width and length". So the height is the smallest dimension, which is \( x - 7 \). So the answer should be B.

Wait, but wait, maybe we made a mistake. Let's check the signs again. If \( x \) is a dimension, it must be positive. So \( x>0 \), \( x + 5>0 \) (always true for \( x>0 \)), and \( x - 7>0 \implies x>7 \). So with \( x>7 \), \( x - 7 \) is positive, and as \( x \) increases, \( x - 7 \) is always less than \( x \) (since we subtract 7) and less than \( x + 5 \) (since \( x - 7 < x < x + 5 \)). Therefore, the smallest dimension is \( x - 7 \), so the height is \( x - 7 \), which is option B.

Answer:

B. \( x - 7 \)